Kohayakawa-Nagle-R{\'o}dl-Schacht conjecture for subdivisions
| Autor: | Chen, Hao, Lin, Yupeng, Ma, Jie |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | In this paper, we study the well-known Kohayakawa-Nagle-R{\"o}dl-Schacht (KNRS) conjecture, with a specific focus on graph subdivisions. The KNRS conjecture asserts that for any graph $H$, locally dense graphs contain asymptotically at least the number of copies of $H$ found in a random graph with the same edge density. We prove the following results about $k$-subdivisions of graphs (obtained by replacing edges with paths of length $k+1$): (1). If $H$ satisfies the KNRS conjecture, then its $(2k-1)$-subdivision satisfies Sidorenko's conjecture, extending a prior result of Conlon, Kim, Lee and Lee; (2). If $H$ satisfies the KNRS conjecture, then its $2k$-subdivision satisfies a constant-fraction version of the KNRS conjecture; (3). If $H$ is regular and satisfies the KNRS conjecture, then its $2k$-subdivision also satisfies the KNRS conjecture. These findings imply that all balanced subdivisions of cliques satisfy the KNRS conjecture, improving upon a recent result of Brada\v{c}, Sudakov and Wigerson. Our work provides new insights into this pivotal conjecture in extremal graph theory. Comment: Add a new lemma (Lemma 3.2) from real analysis |
| Databáze: | arXiv |
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